Tangent from external point

10 The equation of a circle is \(x ^ { 2 } + y ^ { 2 } - 4 x + 6 y - 77 = 0\).

1. Find the \(x\)-coordinates of the points \(A\) and \(B\) where the circle intersects the \(x\)-axis.
2. Find the point of intersection of the tangents to the circle at \(A\) and \(B\).

10 The equation of a circle is \(( x - 3 ) ^ { 2 } + y ^ { 2 } = 18\). The line with equation \(y = m x + c\) passes through the point \(( 0 , - 9 )\) and is a tangent to the circle. Find the two possible values of \(m\) and, for each value of \(m\), find the coordinates of the point at which the tangent touches the circle.
\includegraphics[max width=\textwidth, alt={}, center]{cacac880-5b44-4fae-8ed8-88a095db69cd-16_855_1600_306_233} A function is defined by \(\mathrm { f } ( x ) = \frac { 4 } { x ^ { 3 } } - \frac { 3 } { x } + 2\) for \(x \neq 0\). The graph of \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\) is shown in the diagram.

1. Find the set of values of \(x\) for which \(\mathrm { f } ( x )\) is decreasing.
2. A triangle is bounded by the \(y\)-axis, the normal to the curve at the point where \(x = 1\) and the tangent to the curve at the point where \(x = - 1\). Find the area of the triangle. Give your answer correct to 3 significant figures.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

8 A circle with equation \(x ^ { 2 } + y ^ { 2 } - 6 x + 2 y - 15 = 0\) meets the \(y\)-axis at the points \(A\) and \(B\). The tangents to the circle at \(A\) and \(B\) meet at the point \(P\). Find the coordinates of \(P\).
\includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-10_71_1659_466_244}
\includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-10_2723_37_136_2010}

9 A circle has centre at the point \(B ( 5,1 )\). The point \(A ( - 1 , - 2 )\) lies on the circle.

1. Find the equation of the circle.
   Point \(C\) is such that \(A C\) is a diameter of the circle. Point \(D\) has coordinates (5, 16).
2. Show that \(D C\) is a tangent to the circle.
   The other tangent from \(D\) to the circle touches the circle at \(E\).
3. Find the coordinates of \(E\).

11 A circle with centre \(C\) has equation \(( x - 8 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 100\).

1. Show that the point \(T ( - 6,6 )\) is outside the circle.
   Two tangents from \(T\) to the circle are drawn.
2. Show that the angle between one of the tangents and \(C T\) is exactly \(45 ^ { \circ }\).
   The two tangents touch the circle at \(A\) and \(B\).
3. Find the equation of the line \(A B\), giving your answer in the form \(y = m x + c\).
4. Find the \(x\)-coordinates of \(A\) and \(B\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

9 The line \(y = 2 x + 5\) intersects the circle with equation \(x ^ { 2 } + y ^ { 2 } = 20\) at \(A\) and \(B\).

1. Find the coordinates of \(A\) and \(B\) in surd form and hence find the exact length of the chord \(A B\).
   A straight line through the point \(( 10,0 )\) with gradient \(m\) is a tangent to the circle.
2. Find the two possible values of \(m\).

10. \begin{figure}[h] \end{figure} The circle \(C\) has radius 5 and touches the \(y\)-axis at the point \(( 0,9 )\), as shown in Figure 4.

1. Write down an equation for the circle \(C\), that is shown in Figure 4. A line through the point \(P ( 8 , - 7 )\) is a tangent to the circle \(C\) at the point \(T\).
2. Find the length of \(P T\).

9. \begin{figure}[h] \end{figure} Figure 3 shows a circle \(C\) with centre \(Q\) and radius 4 and the point \(T\) which lies on \(C\). The tangent to \(C\) at the point \(T\) passes through the origin \(O\) and \(O T = 6 \sqrt { } 5\) Given that the coordinates of \(Q\) are \(( 11 , k )\), where \(k\) is a positive constant, (a) find the exact value of \(k\),
(b) find an equation for \(C\).

8. The circle \(C\) has the equation $$x ^ { 2 } + y ^ { 2 } + 10 x - 8 y + k = 0$$ where \(k\) is a constant. Given that the point with coordinates \(( - 6,5 )\) lies on \(C\),

1. find the value of \(k\),
2. find the coordinates of the centre and the radius of \(C\). A straight line which passes through the point \(A ( 2,3 )\) is a tangent to \(C\) at the point \(B\).
3. Find the length \(A B\) in the form \(k \sqrt { 3 }\).

4．The line with equation \(y = m x\) is a tangent to the circle \(C _ { 1 }\) with equation $$( x + 4 ) ^ { 2 } + ( y - 7 ) ^ { 2 } = 13$$ （a）Show that \(m\) satisfies the equation $$3 m ^ { 2 } + 56 m + 36 = 0$$ The tangents from the origin \(O\) to \(C _ { 1 }\) touch \(C _ { 1 }\) at the points \(A\) and \(B\) ．
（b）Find the coordinates of the points \(A\) and \(B\) ．
（8）
Another circle \(C _ { 2 }\) has equation \(x ^ { 2 } + y ^ { 2 } = 13\) ．The tangents from the point \(( 4 , - 7 )\) to \(C _ { 2 }\) touch it at the points \(P\) and \(Q\) ．
（c）Find the coordinates of either the point \(P\) or the point \(Q\) ．
（2）

15. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of a circle \(C\) with centre \(N ( 7,4 )\)
The line \(l\) with equation \(y = \frac { 1 } { 3 } x\) is a tangent to \(C\) at the point \(P\).
Find

1. the equation of line \(P N\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants,
2. an equation for \(C\). The line with equation \(y = \frac { 1 } { 3 } x + k\), where \(k\) is a non-zero constant, is also a tangent to \(C\).
3. Find the value of \(k\).

14. A curve with centre \(C\) has equation $$x ^ { 2 } + y ^ { 2 } + 2 x - 6 y - 40 = 0$$ a. i. State the coordinates of \(C\).
ii. Find the radius of the circle, giving your answer as \(r = n \sqrt { 2 }\).
b. The line \(l\) is a tangent to the circle and has gradient - 7 . Find two possible equations for \(l\), giving your answers in the form \(y = m x + c\).

6. \begin{figure}[h] \end{figure} The circle \(C\) has centre \(A\) with coordinates \(( - 3,1 )\).
The line \(l _ { 1 }\) with equation \(y = - 4 x + 6\), is the tangent to \(C\) at the point \(Q\), as shown in Figure 3.
a. Find the equation of the line \(A Q\) in the form \(a x + b y = c\).
b. Show that the equation of the circle \(C\) is \(( x + 3 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 17\) The line \(l _ { 2 }\) with equation \(y = - 4 x + k , k \neq 6\), is also a tangent to \(C\).
c. Find the value of the constant \(k\).

6. \begin{figure}[h] \end{figure} Not to scale The circle \(C\) has centre \(A\) with coordinates (7,5).
The line \(l\), with equation \(y = 2 x + 1\), is the tangent to \(C\) at the point \(P\), as shown in Figure 3 .

1. Show that an equation of the line \(P A\) is \(2 y + x = 17\)
2. Find an equation for \(C\). The line with equation \(y = 2 x + k , \quad k \neq 1\) is also a tangent to \(C\).
3. Find the value of the constant \(k\).

18. (a) A circle \(C\) has centre \(( - 3 , - 1 )\) and radius \(\sqrt { 5 }\). Show that the equation of \(C\) can be written as \(x ^ { 2 } + y ^ { 2 } + 6 x + 2 y + 5 = 0\).
(b) (i) Find the equations of the tangents to \(C\) that pass through the origin \(O\).
(ii) Determine the coordinates of the points where the tangents touch the circle.
Additional page, if required. Write the question number(s) in the left-hand margin. \section*{PLEASE DO NOT WRITE ON THIS PAGE} \section*{PLEASE DO NOT WRITE ON THIS PAGE} \section*{PLEASE DO NOT WRITE ON THIS PAGE}

8 In this question you must show detailed reasoning. A circle has equation \(x ^ { 2 } + y ^ { 2 } - 6 x - 4 y + 12 = 0\). Two tangents to this circle pass through the point \(( 0,1 )\). You are given that the scales on the \(x\)-axis and the \(y\)-axis are the same.
Find the angle between these two tangents.

11 A circle \(C\) has centre \(( 0,10 )\) and radius \(\sqrt { 20 }\) A line \(L\) has equation \(y = m x\)
11

1. 1. Show that the \(x\)-coordinate of any point of intersection of \(L\) and \(C\) satisfies the equation $$\left( 1 + m ^ { 2 } \right) x ^ { 2 } - 20 m x + 80 = 0$$ 11
2. (ii) Find the values of \(m\) for which the equation in part (a)(i) has equal roots.
   11
3. Two lines are drawn from the origin which are tangents to \(C\). Find the coordinates of the points of contact between the tangents and \(C\).

6 The line \(L\) has equation $$5 y + 12 x = 298$$ A circle, \(C\), has centre \(( 7,9 )\)
\(L\) is a tangent to \(C\).
6

1. Find the coordinates of the point of intersection of \(L\) and \(C\).
   Fully justify your answer.
2. Find the equation of \(C\). 6
3. Find the equation of \(C\).
   \(7 \quad a\) and \(b\) are two positive irrational numbers. The sum of \(a\) and \(b\) is rational. The product of \(a\) and \(b\) is rational.
   Caroline is trying to prove \(\frac { 1 } { a } + \frac { 1 } { b }\) is rational.
   Here is her proof:
   Step \(1 \quad \frac { 1 } { a } + \frac { 1 } { b } = \frac { 2 } { a + b }\)
   Step 22 is rational and \(a + b\) is non-zero and rational.
   Step 3 Therefore \(\frac { 2 } { a + b }\) is rational.
   Step 4 Hence \(\frac { 1 } { a } + \frac { 1 } { b }\) is rational.